数理解析研究所講究録 第 1644 巻 2009 年 86-94 86
Enumeration of Perfect Sequences of Chordal Graph
Yasuko Matsui1, Ryuhei Uehara2, and Takeaki Uno3
1 Department of Mathematical Sciences, School of Science, Tokai University, Kitakaname 1117, Hiratsuka, Kanagawa 259-1292, Japan. [email protected] 2 School of Information Science, JAIST, Asahidai 1-1, Nomi, Ishikawa 923-1292, Japan. uehara@jaist. ac. jp 3 National Institute of Informatics, Hitotsubashi 2-1-2, Chiyoda-ku, Tokyo 101-8430, [email protected]
Abstract. A graph is chordal if and only if it has no chordless cycle of length more than three. The set of maximal cliques in a chordal graph admits special tree structures called clique trees. A perfect sequence is a sequence of maximal cliques obtained by using the reverse order of repeatedly removing the leaves of a clique tree. This paper addresses the problem of enumerating all the perfect sequences. Although this problem has statistical applications, no efficient algorithm has been proposed. There are two difficulties with developing this type of algorithms. First, a chordal graph does not generally have a unique clique tree. Second, a perfect sequence can normally be generated by two or more distinct clique trees. Thus it is hard using a straightforward way to generate perfect sequences from each possible clique tree. In this paper, we propose a method to enumerate perfect sequences without constructing clique trees. As a result, we have developed the first polynomial delay algorithm for dealing with this problem. In particular, the time complexity of the algorithm on average is $O(1)$ for each perfect sequence. Keywords: chordal graph, clique tree, enumeration, perfect sequence.
1 Introduction
A graph is said to be chordal if every cycle of length at least 4 has a chord. Chordal graphs have been investigated for a long time for many areas. From the viewpoint of graph theory, this class has a simple characterization; a graph is chordal if and only if it is an intersection graph of the subtrees of a tree. That
is, there is a set of subtrees $T_{v}$ of a tree $\mathcal{T}$ that correspond to the vertices $v$ of a chordal graph $G$ such that
$u$ and $v$ are adjacent in $G$ if and only if the corresponding subtrees $T_{u}$ and $T_{v}$ have non-empty intersection. From an algorithmic point of view, a chordal graph is characterized by a simple vertex ordering called a perfect elimination ordering (PEO). Its geometrical property of rigidity plays an important role in many practical areas, and the property of its adjacency matrix is useful in matrix manipulations. Since the class appears in different contexts in so many areas they also are called “rigid circuit graphs” or “triangulated graphs” (see e.g., [13, 5]). Recently, the probabilistic relationships between random variables are represented by a graph called ”graphical model” in the science of statistics. On a graphical model, the random variables are represented by the vertices and the conditional dependencies are represented by the edges. In particular, if a graphical model is chordal, we can easily compute its maximum likelihood estimator. In the computation, a chordal graph is decomposed into its subgraphs by removing a separator which induces a clique. Therefore, chordal graphs are also called “decomposable graphs” or “decomposable models” in this area (see, e.g., [9]). Even if a given graphical model is not decomposable, we sometimes add edges to make it decomposable to obtain a good approximation value of the maximum likelihood estimator $oi$ the graphical model (see,
e,g., [4] $)$ . This corresponds to the notion of ”chordal completion” in the area of graph algorithms [8]. Hence chordal graphs play an important role in graphical modeling in statistics.
Perfect sequences are the sequences of maximal cliques in a given chordal graph that $Satis\mathfrak{h}r$ certain properties. The notion arises fiom the decomposable models, and all perfect sequences are required and 87
must not have repetitions (see, e.g., [7]) From the viewpoint of graph theory, perfect sequences can be seen in the following way. As mentioned, a chordal graph $G$ can be represented by the intersection graph
$\mathcal{T}$ of the subtrees of a tree . That is, each vertex $v$ of $G=(V, E)$ corresponds to a subtree $T_{v}$ of $\mathcal{T}$ , and
$T_{v}$ $T_{u}$ $\mathcal{T}$ $\{u, v\}\in E$ if and only if and intersect. We can make each node $c_{1}$ of tree correspond to a maximal
$C_{i}$ $G;C_{j}$ $G$ $T_{v}$ clique of consists of all the vertices $v$ in such that contains the node $c_{i}$ . Therefore, the tree
$\mathcal{T}$ $I^{l}\neg rom$ $\mathcal{T}$ is called a clique tree of G. the clique tree , we make an ordering $\pi$ over the set of maximal
$\mathcal{T}$ cliques $\{C_{1}, C_{2}, \ldots , C_{k}\}$ of $G$ such that $C_{\pi(i)}$ is a leaf of tree $\mathcal{T}_{i}$ which is a subgraph of induced by
$C_{\pi(1)},$ $C_{\pi(2)},$ $C_{\pi(l)}$ $i$ $\mathcal{T}$ $\ldots,$ for each . Intuitively, we cari construct such a sequence from by repeatedly pruning leaves and put them on the top of the sequence until $\mathcal{T}$ is empty. Then the sequence of maximal cliques in a chordal graph is called a perfect sequence, and it is known that a graph is chordal if and only if it has a perfect sequence. In 2006, Hara and Takemura proposed a sampling algorit}im for the perfect sequences of a given chordal graph [6] that used the Lauritzen’s method [9]. However their algorithm does not generate each perfect sequence uniformly at random, and to our knowledge, no enumeration algorithm of perfect sequences exists. There are two major reasons for the difficulty in enurnerating perfect sequences. First, the clique tree is not generally unique for a chordal graph. That is, a chordal graph has many distinct (non- isomorphic) clique trees in general. For a clique tree, we can define a set of perfect sequences consistent to the clique tree. Then, secondly, the sets of perfect sequences consistent with the distinct clique trees are not disjoint. That is, we can obtain one perfect sequence from possibly many distinct clique trees. Therefore, a straightforward algorithm based on a simple idea (generate all clique trees, and generate all perfect sequences for each clique tree) cannot avoid redundancy. In this paper, we propose an algorithm enumerating all the perfect sequences of a chordal graph. The algorithm enumerates all the perfect sequences on an average of $O(1)$ time per sequence. In order to avoid redundancy, our algorithm makes a weighted intersection graph of maximal cliques first instead of explicitly constructing the clique trees. The intersection graph is uniquely constructed, and each maxi- mum $weig\}ited$ spanning tree of the intersection graph gives a clique tree of a chordal graph. Then the algorithm generates each perfect sequence from the union of maximum weighted spanning trees without any repetitions. The algorithm is based on a new idea that characterizes the union of maximum weighted spanning trees, and that also gives us insight into the properties of the set of clique trees of a chordal graph. We note that the set of perfect sequences is strongly related to the set of PEOs. The PEO is a standard characterization of a chordal graph in the area of graph algorithms. Any PEO can be obtained by repeatedly removing a simplicial vertex, and any sequence of removals of simplicial vertices is a PEO. This property admits us to enumerate all PEOs by recursively removing simplicial vertices. The enumeration of all PEOs is investigated by Chandran et al. [3]. Intuitively, any perfect sequenoe can be obtained by removing a set of “equivalent” simplicial vertices together repeatedly. However, each removal has to follow (a kind of) lexicographical ordering. A removal of a set of simplicial vertices may produce a new set of simplicial vertices or change another set of simplicial vertices, and the new set cannot be chosen before the old sets. Thus the family of sets of equivalent simplicial vertices has to follow a partial ordering defined by the appearance in the graph to obtain a perfect sequence. This fact also implies that while some PEOs can be obtained from a perfect sequence easily, but some PEOs camiot be obtained from any perfect sequence straightforwardly by the constraint of the partial ordering. Therefore, tbe correspondence between the set of perfect sequences and the set of PEOs is not straightforward and hence we have to analyze some special cases. Indeed, using this approach, we can obtain another enumeration algorithm of all perfect sequences from the enumeration algorithm of all PEOs. However this approach does not allow us to enumerate efficiently; the algorithm takes $O(|V|+|E|)$ time for each perfect sequence. This 88
is the reason why we take completely different approach based on a maximum weighted spanning tree of a weighted clique graph, which admits us to improve the time to $O(1)$ on average.
2 Preliminaries
The neighborhood of a vertex $v$ in a graph $G=(V, E)$ is the set $N_{G}(v)=\{u\in V|\{u\rangle v\}\in E\}$ , and the degree of a vertex $v$ is $|N_{G}(v)|$ and is denoted by $\deg_{G}(v)$ . For a vertex subset $U$ of $V$ we denote by $N_{G}(U)$ the set $\{v\in V|v\in N_{G}(u)$ for some $u\in U\}$ . If no confusion arises we will omit the subscript $G$ . Given a graph $G=(V, E)$ and a subset $U\subseteq V$ , the subgmph of $G$ induced by $U$ is the graph $(U, F)$ , where $F=\{\{u, v\}\in E|u, v\in U\}$ , and denoted by $G[U]$ . A vertex set $I$ is an independent set of $G$ if $G[I]$ contains no edge, and a vertex set $C$ is a clique in $G$ if any pair of vertices in $C$ is connected by an edge in $G$ . An edge $e$ in a connected graph $G=(V, E)$ is called a bridge if its removal partitions $G$ into two connected components.
For a given graph $G=(V, E)$ , consider a sequence $(v_{0}, v_{1}, \ldots, vp)$ of vertices in $V$ such that $\{v_{j-1}, v_{j}\}\in$ $E$ $0 $v_{0},$ if the vertices $\ldots,$ $v_{\ell-1}$ are distinct and $v_{0}=v_{\ell}$ . The length of such a path and a cycle is the number $\ell$ . $A_{I1}$ edge that joins the two vertices of a cycle, but is not itself an edge of the cycle, is a chord of the cycle. A graph is chordal if each cycle of length at least 4 has a chord. Given a graph $G=(V, E)$ , a vertex $v\in V$ is simplicial in $G$ if $N(v)$ is a clique in $G$ . An ordering $v_{1},$ $v_{n}$ $V$ $G$ $\ldots$ , of the vertices of is a perfect elimination ordenng of if the vertex $v_{i}$ is simplicial in $G[\{v_{i}, v_{i+1}, \ldots, v_{n}\}]$ $n$ for all $i=1,$ $\ldots,$ . It is known that a graph is chordal if and only if it has a perfect elimination ordering [2, Section 1.2]. Given a chordal graph, a perfect elimination ordering of the graph can be found in linear time [12, 14]. For a chordal graph $G=(V, E)$ , we can associate a tree $\mathcal{T}$ , called a clique tree of $G$ , as satisfying the $\mathcal{T}$ following three properties. (A) The nodes4 of are the maximal cliques of G. (B) Two nodes of $\mathcal{T}$ are adjacent only if the intersection of the corresponding maximal cliques is non-empty. (C) For every vertex $G$ $\mathcal{T}$ $v$ of , the subgraph $T_{v}$ of induced by the maximal cliques containing $v$ is connected. (Here, condition (A) is sometimes weakened as each node is not necessarily maximal.) It is well known that a graph is chordal if and only if it has a clique tree, and in such a case a clique tree can be constructed in linear $V$ $\mathcal{T}$ time. On the tree, each vertex $v$ in corresponds to a subtree $T_{v}$ of . That is, $T_{v}$ consists of maximal cliques that contain $v$ . Then, the graph $G$ is an intersection graph of subtrees $T_{v}$ of a tree $\mathcal{T}$ . Some of these details are explained in books [2, 13]. For a given chordal graph $G=(V, E)$ , we denote the set of all maximal cliques of $G$ by $C(G)$ . (It is known that $|C(G)|\leq|V|.)$ Let $k=|C(G)|,$ $C(G)=\{C_{1}, C_{2}, \ldots, C_{k}\}$ , and $\pi$ be a permutation of $k$ $C_{\pi(1)},$ $C_{\pi(2)},$ $C_{\pi(k)}$ $C(G)$ elements. Then, the ordering $\ldots,$ on is said to be a perfect sequence if there is a $\mathcal{T}$ clique tree such that each $C_{n(i)}$ is a leaf of the subtree $\mathcal{T}[\{C_{\pi(1)}, C_{\pi(2)}, \ldots , C_{\pi(i)}\}]$ which is the subgraph $\mathcal{T}$ of induced by $\{C_{\pi(1)}, C_{\pi(2)}, \ldots, C_{\pi(i)}\}$ , for each $i$ with $1\leq i\leq k$ . Intuitively, we have two explanations for this. One is that we can prune all the leaves off a clique tree in the reverse order $C_{\pi(k)’}C_{\pi(2)},$ $C_{\pi(1)}$ any perfect $C_{\pi(1)},$ $C_{\pi(2)},$ $C_{\pi(k)}$ of sequence. On the other hand, according to the perfect sequence $\ldots,$ , we $\mathcal{T}$ can constmct a clique tree by repeatedly attaching $C_{\pi(i)}$ as a leaf. We here note that, in general, a clique tree for a chordal graph $G$ is not uniquely determined up to isomorphism. For example, for the chordal graph $G=(V, E)$ in Fig. 1(a), there are four distinct clique trees given in Fig. 1(b). Moreover, two or more distinct clique trees of the same chordal graph $G$ can generate the same perfect sequence. For example, for the chordal graph $G=(V, E)$ in Fig. 1(a), a perfect sequence $C_{2}C_{1}C_{3}C_{4}C_{5}$ can be generated from two trees, $T_{1}$ and $T_{2}$ , as depicted in Fig. 1 (b)(c). 4 In this paper, “vertex“ is in a chordal graph $G$ , and “node” corresponds to a clique in $G$ to distinguish them. 89 (a) Chordal graph $G=(V, E)$ (d) Weighted clique graph $cq(G)$ . Fig. 1. (a) Cbordal graph $G,$ $(b)$ its clique trees, (c) corresponding perfect sequences, and (d) its weighted clique $graI)1_{1}C\mathcal{G}(G)=(C(G),\mathcal{E})$ For a chordal graph $G=(V, E)$ and the set $C(G)$ of all maximal cliques, we define the weighted clique $C\mathcal{G}(G)=(C(G), \mathcal{E})$ $\mathcal{E}arrow \mathbb{Z}$ graph with a weight function $w$ : as follows. For two maximal cliques $C_{1}$ $C_{2}$ $\mathcal{E}$ $C(G),$ $\{C_{1}, C_{2}\}$ $C_{1}\cap C_{2}\neq\emptyset$ $\mathcal{E}$ and in contains the edge if and only if . For each edge $\{C_{1}, C_{2}\}$ in , $w(\{C_{1}, C_{2}\})$ $\mathcal{E}$ is defined by $|C_{1}\cap C_{2}|$ (therefore every edge in has a positive integer weight less than $|V|$ ). The $C\mathcal{G}(G)=(C(G), \mathcal{E})$ of a chordal graph $G=(V, E)$ in Fig. 1 (a) is given in Fig. 1(d). The weights of the edges are all 1 except for $\{C_{2}, C_{3}\}$ which has a weight 2. We recall that each edge in a clique tree $\mathcal{T}$ of $G$ corresponds to a nonempty intersection of two maximal $\mathcal{T}$ cliques. Thus, is a spanning tree of $C\mathcal{G}(G)$ . However, some spanning trees of $C\mathcal{G}(G)$ may not be clique trees of $G$ . The characterization of a clique tree is given as follows. Lemma 1 (e.g., [1, 11]). Let $G=(V, E)$ be a chordal graph and $C\mathcal{G}(G)=(C(G),\mathcal{E})$ be the weighted clique graph with a weight function $w.$ A spanning tree $\mathcal{T}$ of $C\mathcal{G}(G)$ is a clique tree of $G$ if and only if it has the maximum weight. For the $C\mathcal{G}(G)=(C(G), \mathcal{E})$ in Fig. 1 (d) of the chordal graph $G=(V, E)$ in Fig. 1 (a), the only spanning trees that contain the edge $\{C_{2}, C_{3}\}$ are clique trees of $G$ . We note that any chordal graph of $n$ vertices contains $n$ maximal cliques at the most. Therefore, $C\mathcal{G}(G)$ contains $O(n)$ nodes. On the other hand, although a star $S_{n}$ of $n$ vertices contains $n$ vertices, $n-1$ edges, and $n-1$ maximal cliques, the clique graph $C\mathcal{G}(S_{n})$ is a complete graph $K_{n-1}$ with $n-1$ nodes that contains $(\begin{array}{l}n-12\end{array})=O(n^{2})$ edges. Therefore, 90 $O(|V|^{2})$ we only have a trivial upper bound for the number of edges in the clique graph $C\mathcal{G}(G)$ of a chordal graph $G=(V, E)$ , even if $|E|=O(|V|)$ . Hereafter, we assume that the input graph $G$ is connected without loss of generality; in case $G$ is not connected, we allow the use of $h$ weight-zero edges to join the $h$ connected components. The modiflcation of the algorithms in this paper is straightforward, and omitted. We show here a technical lemma for a clique tree that will be referred to later: Lemma 2. Suppose $\mathcal{T}$ is a clique tree of a chonlal gruph $G=(V, E)$ that consists of at least two maximal cliques (in the other wods, $G$ is not a complete graph). Let $C$ be a leaf in $\mathcal{T}$ and $C’$ be the unique neighbor of C. Then for each vertex $v$ in $C,$ $v$ is simplicial in $G$ if and only if $v$ is in $C\backslash C’$ . $\square$ 3 Enumeration Algorithm The idea for enumerating perfect sequences is simple. We construct a graph representing the adjacency of maximal cliques, and recursively remove the maximal cliques that can be leaves of a clique tree. Since the clique tree is a spanning tree of $t\}_{1}e$ graph and the removed maximal clique is a leaf of the clique tree, after removing the maximal cliques, we still have a clique (sub)tree that is a spanning tree of the resultant graph. Since any tree has at least two leaves, we always have at least two maximal cliques that correspond to the leaves of the spanning tree. Thereforc, we invariably get a perfect sequence by repeating the removal process. During the algorithm, the spanning tree is not explicitly given, and we have to deal with all the potential spanning trees that can generate perfect sequences. The following outlines a description of the algorithm. $\frac{A1gor1thm1:Out1ineofEnumeration}{Input:C1_{1}orda1graphG=(V,E);}$ Output: All perfect sequences of $G$ : 1 construct weigbted clique graph $C\mathcal{G}(G)$ ; 2 conipute arbitrary maximum weighted spanning tree $\mathcal{T}^{*}$ of $C\mathcal{G}(G)$ ; a construct graph $C\mathcal{G}(G)^{*}$ composed of edges that can be included in clique trees from $C\mathcal{G}(G)$ and $\mathcal{T}^{*}$ ; 4 $e\iota iuirierate$ all sequences of maximal cliques obtained by repeatedly removing maximal cliques that can be leaves of sorne clique trees. The graph $C\mathcal{G}(G)^{*}$ is a subgraph of $C\mathcal{G}(G)$ that excludes unnecessary edges described later. To efficiently find the maximal cliques that can be leaves, we first compute any maximum weighted $\tau*$ spanning tree . Then, we use a (unweighted) graph $C\mathcal{G}(G)^{*}$ obtained from $C\mathcal{G}(G)$ and $\tau*$ . We say an edge in $C\mathcal{G}(G)$ is unnecessary if it cannot be included in any maximum weighted spanning tree of $C\mathcal{G}(G)$ . On the other hand, an edge is indispensable if it appears in every maximum weighted spanning tree of $C\mathcal{G}(G)$ . The other edges are called dispensable, which means they appear in some (but not all) maximum $\mathcal{T}$ $\mathcal{T}$ “ weighted spanning trees. Let $e$ be an edge not in . Since is a spanning tree of $C\mathcal{G}(G)$ , the addition $\tau*$ $\mathcal{T}^{*}$ of $e$ to produces a unique cycle $C_{\epsilon}$ which consists of $e$ and the other edges in . We call C. an elementary cycle of $e$ (the terminology comes from the matroid theory). The unnecessary, dispensable, and indispensable edges are characterized by the following lemmas. Lemma 3. For an edge $e$ not in $\mathcal{T}_{f}^{*}w(e)\leq w(e’)$ holds for any $e’\in C_{e}\backslash \{e\}$ . Moreover, $e$ is unnecessary if and only if $w(e) “ Lemma 4. An edge $e$ in $\mathcal{T}$ is an indispensable edge if $w(e)>w(e’)$ for all edges $e’$ such that $e’$ is not on $\tau*andC_{e}/containse$ . Proof is analogous to proof of Lemma 3, and hence omitted. $\mathcal{T}$ “ We note that any bridge is indispensable by Lemma 4; there is no edge $e’$ not in such that $C_{e}/$ contains $e$ and $w(e’)\geq w(e)$ . We denote the sets of unnecessary, indispensable, and dispensable edges by $\mathcal{E}_{u},$ $\mathcal{E}_{i}$ , and $\mathcal{E}_{d}$ , respectively. They partition the edge set $\mathcal{E}$ of $C\mathcal{G}(G)$ into three disjoint sets. The sets can be computed by the following algorithm in $O(|C(G)|^{3})=O(|V|^{3})$ time. We note that by using a dynamic programming technique starting from the bottom of the tree, the mn time can be reduced to $O(|C(G)|^{2})$ , which is omitted here since it is too complex and tedious. $\overline{\frac{A1orithm2:Searc\}_{1}forUm1ecessary,Indi_{S}ensab1e,andDiensa\dagger_{J}1eFdges}{Input:Thev^{r}eightedcliquegraphC\mathcal{G}(G)=(C(G),\mathcal{E})andanarbitrary\max imumweig\}ited}}$ sparming tree $\mathcal{T}^{*}$ of $C\mathcal{G}(G)$ ; Output: Sets $\mathcal{E}_{u},$ $\mathcal{E}_{i}$ , and $\mathcal{E}_{d}$ of the unnecessary, indispensable, and dispensable edges; 1 set $\mathcal{E}_{u}:=\emptyset;\mathcal{E}_{d}:=\emptyset;\mathcal{E}_{i}:=\emptyset$ ; $\mathcal{T}$ “ 2 $e$ not in $w(e) 3 $\mathcal{E}_{d}:=\mathcal{E}_{d}\cup\{e\}$ ; 4 $e’\in C_{c}$ satisfyirig $w(e)=w(e’)\mathcal{E}_{d}:=\mathcal{E}_{d}\cup\{e’\}$ ; $\epsilon \mathcal{E}_{i}:=E\backslash (\mathcal{E}_{u}\cup \mathcal{E}_{d})$ ; 6 $(\mathcal{E}_{i}, \mathcal{E}_{u}, \mathcal{E}_{d})$ : We now define an unweighted graph $C\mathcal{G}(G)^{*}$ by $(C(G), \mathcal{E}_{i}\cup \mathcal{E}_{d})$ . Observation 1 Any spanning tree of $C\mathcal{G}(G)^{*}$ that contains all the edges in $\mathcal{E}_{i}$ gives a maximum weighted spanning tree of $C\mathcal{G}(G)$ . Now, we have characterized clique trees by using spanning trees in $C\mathcal{G}(G)$ . Next, we take the charac- terization of maximal cliques that can be leaves of some clique trees into consideration. Lemma 5. A maximal clique $C$ can be a leaf of a clique tree if and only if $C$ satisfies (1) $C$ is incident to at most one edge in $\mathcal{E}_{i}$ , and (2) $C$ is not a cut vertex in $C\mathcal{G}(G)^{*}$ . Proof. First, we suppose that $C$ is a leaf of a clique tree $\mathcal{T}$ . Sinoe $\mathcal{T}$ is a clique tree of $G,$ $\mathcal{T}$ is a spanning “ tree in $C\mathcal{G}(G)$ that includes all the edges in $\mathcal{E}_{i}$ . Since $C$ is a leaf of $\mathcal{T},$ $C$ is incident to at most one edge $\mathcal{E}_{i}$ $C$ of , and is not a cut vertex of $C\mathcal{G}(G)^{*}$ . Thus, $C$ satisfies the conditions. We next suppose that $C$ satisfies the conditions. We assume that $C\mathcal{G}(G)$ contains two or more nodes. $\mathcal{E}_{i}\cup \mathcal{E}_{d}$ We choose any edge $e$ from that is incident to $C$ . We always can choose $e$ since $C\mathcal{G}(G)$ is connected. Then, we remove $C$ from $C\mathcal{G}(G)^{*}$ . Since $C$ is not a cut vertex, the resultant graph $C\mathcal{G}(G)’$ is still connected. $C\mathcal{G}(G)$ ‘ $\mathcal{T}’$ Therefore, has a spanning tree which contains all the edges in $\mathcal{E}_{i}\backslash \{e\}$ . Then, by adding $e$ to $\mathcal{T}^{l}$ $\mathcal{T}$ $\mathcal{T}$ , we have a spanning tree that contains all the edges in $\mathcal{E}_{i}$ , and $C$ is a leaf of . This concludes the proof. $\square$ Hereafter, the pair of two conditions in Lemma 5 is said to be a leaf condition. A perfect sequence is obtained by removing a leaf of a clique tree $\mathcal{T}$ . Thus, any perfect sequenoe is obtained by iteratively removing the maximal cliques satisfying a leaf condition. The converse is shown by the following lemma. Lemma 6. Any maximal clique sequence $S=(C_{1}, \ldots, C_{k})$ obtained by iteratively removing a maximal clique satisfying the leaf condition is a perfect sequence. 92 $\mathcal{T}_{1}$ Proof. Let be a tree in $C\mathcal{G}(G)^{*}$ that consists of one vertex $C_{1}$ , and $\mathcal{T}_{j}$ be a tree in $C\mathcal{G}(G)^{*}$ obtained $C_{i}$ $\mathcal{T}_{i-1}$ by adding to and an edge $e$ of $C\mathcal{G}(G)^{*}$ connecting $C_{i}$ and a vertex in $\mathcal{T}_{i-1}$ . If there is an edge of $\mathcal{E}_{i}$ $C_{i}$ $\mathcal{T}_{i-1}$ connecting and a vertex in , we choose the edge as $e$ . Note that there is at most one such edge $C_{i}$ $\mathcal{T}_{i}$ by the leaf condition. We observe that any is a leaf of , and $\mathcal{T}_{i-1}$ is obtained by removing a leaf $C_{i}$ $\mathcal{T}_{i}$ from . Thus, $S=(C_{1}, \ldots, C_{k})$ is a perfect sequence. $\square$ This lemma ensures that by repeatedly removing maximal cliques satisfying the leaf condition, we can obtain any perfect sequence. This yields the following algorithm to enuinerate all perfect sequences. $\overline{Algorithm3:}$All Perfect Sequences :Chordal $grapliG=(V, E)$ ; $Output:\overline{Input}$All perfect sequences of $G$ : 1 construct $C\mathcal{G}(G)$ ; 2 find maximum weighted spanning tree $\mathcal{T}^{*}$ of $C\mathcal{G}(G)$ ; $\mathcal{T}^{*}$ a by using , compute sets $\mathcal{E}_{u},$ $\mathcal{E}_{i},$ $\mathcal{E}_{d}$ of unnecessary, indispensable, and dispensable edges, respectively; $P$ 4 set to empty sequence $*[r]$ keep current perfect sequence let $C\mathcal{G}(G)^{*}$ $:=(C(G), \mathcal{E}_{i}\cup \mathcal{E}_{d})$ ; $\epsilon$ call Enumerate $(C\mathcal{G}(G)^{*}, P)$ ; $\overline{\frac{ProcedureEnumerate(C\mathcal{G}(G)^{*}=(C(G),\mathcal{E}_{i}\cup \mathcal{E}_{d}),P)}{Output:Aperfectseq_{11}enceatt1_{1}e1astnode;}}$ $\tau C(G)$ contains one node $C$ output $(C+P)^{*}[r]C+P$ denotes concatenation of node $C$ and sequence $P$ compute $S:=$ { $C\in C(G)|C$ satisfies the leaf condition}; $sC\in S$ call Enumerate $(C\mathcal{G}(G)\backslash C, C+P)$ ; A part of the computation tree for the chordal graph in Fig. 1 (a) is given in Fig. 2. The unweighted graph $C\mathcal{G}(G)^{*}$ contains two indispensable edges, $\{C_{2}, C_{3}\}$ and $\{C_{4}, C_{5}\}$ . Now we are ready to show the main theorem in this paper. Theorem 1. For any chordal graph $G=(V, E)$ , with $O(|V|^{3})$ time and $O(|V|^{2})$ space pre-computation, all perfect sequences can be enumerated in $O(1)$ time per sequence on average and $O(|V|^{2})$ space. Proof. From Lemmas 5 and 6, we can see $t$ }$iat$ Algorithm 3 generates all perfect sequences. Since each iteration adds a maximal clique on the top of the sequenoe when it generates a recursive call, no two iterations can output the same perfect sequence. Thereby any perfect sequence is generated exactly once. This shows the correctness of the algorithm. Therefore, we concentrate on the analysis of the time complexity. The space complexity is easy to see. We first observe that the computation of set $S$ in step 10 takes $O(n^{2})$ time, where $n=|C(G)|$ in the procedure. Therefore, the time complexity of each procedure call of Enumerate can be bounded above by 2 en time for a positive constant $c$ except for the computation time spent for the recursive calls generated by the procedure. Now, a procedure call of Enumerate where $C\mathcal{G}(G)=(C(G), \mathcal{E}_{i}\cup \mathcal{E}_{d})$ is called a k-level call if $|C(G)|=k$ . Let $t(k)$ be the total computation time for k-level calls. When $k=1$ , we have a perfect sequence for each call. Therefore, we have $t(1)\leq cN$ , where $N$ is the number of perfect sequences. When $k>1$ , there are at least two maximal cliques in $C(G)$ satis $\mathfrak{h}ring$ the leaf condition. Therefore, there are at least two cliques in $S$ , and the number of k-level calls is at most half of the number of $(k-1)-$ level calls. Thus, we have $t(k)\leq\overline{2}^{P\neg}k^{2}-cN$ if $k>1$ . We can see that $t(2)+t(3)+t(4)\leq(2+9z+2)cN=T25_{cN}$ . For any $k>4$ , we have $\overline{2}^{T-T}k^{2}\leq\circ 8\frac{(k-1)^{2}}{2^{\kappa-s}}$. Thus, we have $\sum_{k=5}^{\infty}t(k)\leq(_{\S}^{8}+(_{\S}^{8})^{2}+\cdots)t(4)<8t(4)\leq 16cN$ . Therefore, each perfect sequence can be obtained on average in $O(1)$ time. $\square$ 93 Fig. 2. Part of computation tree that enumerates all perfect sequences We note here that we can include the time to output in $O(1)$ time for each perfect sequence on average. The first idea is to output the difference of the previous output. The second idea is to output at each step when the clique of the sequence is found. More precisely, we replace step 8 of Enumerate by the following three steps output $+C$ ; output “end of reverse of a perfect sequence” ; output $-C$ ; and replace step 12 of Enumerate by the following three steps output $+C$ ; call Enumerate $(C\mathcal{G}(G)\backslash C, C+P)$ ; output $-C$ ; Then we will incrementally have all perfect sequences, and the time complexity is still $O(1)$ time for each perfect sequence on average. We also note that the delay in Algorithm 3, which is the maximum time between two consecutive perfect sequences, is $O(|V|^{3})$ with a straightforward implementation. This time complexity comes from (1) the distance between two perfect sequences in the computation tree is $O(n)$ and (2) each computation 94 of the set $S$ takes $O(n^{2})$ time in step 10. However, we do not need to compute $S$ completely every time. We can update $S$ incrementally for each removal and addition with a suitable data structure, and the computation time can be reduced to $O(n)$ time. Hence it is not difficult to reduce the delay to $O(n^{2})$ , but the details are omitted here. 4 Conclusion We propose an algorithm to enurnerate all the perfect sequences of a given chordal graph. The time complexity for each perfect sequence is $O(1)$ , which is the optimal time complexity. From the proof of the main theorem, we can see that the number of perfect sequences might be exponential in the size of the graph in general, and thus for large graphs the algorithm is impractical. Therefore, one of our future works is to construct an efficient random sampling algorithm. Our approach does not use clique trees, and thus, with polynomial time convergence, there is the possibility for efficient sampling, References 1. J. R. S. Blair and B. Peyton. An Introduction to Chordal Graphs and Clique Trees. 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